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A Regular Countable Chain Condition Space without Compact‐caliber (ω 1 , ω)
Author(s) -
MCINTYRE D. W.
Publication year - 1993
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1993.tb52528.x
Subject(s) - countable set , disjoint sets , mathematics , chain (unit) , space (punctuation) , regular space , second countable space , caliber , compact space , set (abstract data type) , topology (electrical circuits) , topological space , cosmic space , discrete mathematics , combinatorics , pure mathematics , computer science , physics , materials science , metallurgy , programming language , operating system , astronomy
. A topological space has the countable chain condition (CCC) if every disjoint collection of nonempty open sets is countable. It has compact‐caliber (ω 1 , ω) if, for every family { Uα : α∈ω 1 ) of nonempty open sets, there is a compact set K such that K ∩ U α |Mn Ø for infinitely many α∈ω 1 . It has been previously shown that CCC implies compact‐caliber (ω 1 , ω) for first countable regular spaces. An example is constructed to show that CCC does not imply compact‐caliber (ω 1 , ω) for arbitrary regular spaces. The method of construction is to refine the usual topology on the set of real numbers, and take the Pixley‐Roy space over this refinement.